p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.461C23, C4.412+ 1+4, (C8×D4)⋊16C2, D4.Q8⋊5C2, C8⋊8D4⋊35C2, C4⋊C8⋊70C22, C4⋊C4.260D4, (C4×C8)⋊46C22, (C4×SD16)⋊38C2, D4.7D4⋊9C2, (C2×D4).233D4, D4⋊5D4.2C2, Q8.D4⋊7C2, (C2×Q16)⋊8C22, (C4×Q8)⋊25C22, C4.Q8⋊37C22, C2.D8⋊11C22, D4.17(C4○D4), C22⋊SD16⋊35C2, C8.18D4⋊11C2, C4⋊C4.227C23, C22⋊C8⋊63C22, (C2×C4).488C24, (C2×C8).347C23, (C22×C8)⋊40C22, C22⋊C4.100D4, C22.9(C4○D8), C23.106(C2×D4), C22⋊Q8⋊16C22, C42.C2⋊7C22, C2.67(D4○SD16), Q8⋊C4⋊49C22, (C4×D4).330C22, (C2×D4).420C23, C4⋊D4.71C22, C23.19D4⋊4C2, (C2×Q8).205C23, C2.124(D4⋊5D4), C42⋊C2⋊20C22, C23.24D4⋊26C2, C23.47D4⋊31C2, C4.4D4.58C22, C22.748(C22×D4), D4⋊C4.118C22, C22.46C24⋊1C2, (C22×C4).1132C23, (C2×SD16).154C22, (C22×D4).407C22, C42.78C22⋊17C2, C2.58(C2×C4○D8), C4.213(C2×C4○D4), (C2×C4).165(C2×D4), (C2×D4⋊C4)⋊26C2, C22⋊C4○(D4⋊C4), (C2×C4⋊C4).658C22, (C2×C4○D4).196C22, SmallGroup(128,2028)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.461C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >
Subgroups: 432 in 204 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C22×C8, C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.24D4, C8×D4, C4×SD16, C22⋊SD16, D4.7D4, Q8.D4, C8⋊8D4, C8.18D4, D4.Q8, C23.19D4, C23.47D4, C42.78C22, D4⋊5D4, C22.46C24, C42.461C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C4○D8, D4○SD16, C42.461C23
►On 32 points
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 28 24 19)(2 25 21 20)(3 26 22 17)(4 27 23 18)(5 12 15 31)(6 9 16 32)(7 10 13 29)(8 11 14 30)
(1 10 22 31)(2 32 23 11)(3 12 24 29)(4 30 21 9)(5 19 13 26)(6 27 14 20)(7 17 15 28)(8 25 16 18)
(1 22)(2 23)(3 24)(4 21)(5 29)(6 30)(7 31)(8 32)(9 14)(10 15)(11 16)(12 13)(17 19)(18 20)(25 27)(26 28)
(1 23 24 4)(2 3 21 22)(5 14 15 8)(6 7 16 13)(9 10 32 29)(11 12 30 31)(17 25 26 20)(18 19 27 28)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,10,22,31)(2,32,23,11)(3,12,24,29)(4,30,21,9)(5,19,13,26)(6,27,14,20)(7,17,15,28)(8,25,16,18), (1,22)(2,23)(3,24)(4,21)(5,29)(6,30)(7,31)(8,32)(9,14)(10,15)(11,16)(12,13)(17,19)(18,20)(25,27)(26,28), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,10,22,31)(2,32,23,11)(3,12,24,29)(4,30,21,9)(5,19,13,26)(6,27,14,20)(7,17,15,28)(8,25,16,18), (1,22)(2,23)(3,24)(4,21)(5,29)(6,30)(7,31)(8,32)(9,14)(10,15)(11,16)(12,13)(17,19)(18,20)(25,27)(26,28), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,28,24,19),(2,25,21,20),(3,26,22,17),(4,27,23,18),(5,12,15,31),(6,9,16,32),(7,10,13,29),(8,11,14,30)], [(1,10,22,31),(2,32,23,11),(3,12,24,29),(4,30,21,9),(5,19,13,26),(6,27,14,20),(7,17,15,28),(8,25,16,18)], [(1,22),(2,23),(3,24),(4,21),(5,29),(6,30),(7,31),(8,32),(9,14),(10,15),(11,16),(12,13),(17,19),(18,20),(25,27),(26,28)], [(1,23,24,4),(2,3,21,22),(5,14,15,8),(6,7,16,13),(9,10,32,29),(11,12,30,31),(17,25,26,20),(18,19,27,28)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | D4○SD16 |
| kernel | C42.461C23 | C2×D4⋊C4 | C23.24D4 | C8×D4 | C4×SD16 | C22⋊SD16 | D4.7D4 | Q8.D4 | C8⋊8D4 | C8.18D4 | D4.Q8 | C23.19D4 | C23.47D4 | C42.78C22 | D4⋊5D4 | C22.46C24 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.461C23 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 14 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,0,14,0,0,11,0],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4] >;
C42.461C23 in GAP, Magma, Sage, TeX
C_4^2._{461}C_2^3 % in TeX
G:=Group("C4^2.461C2^3"); // GroupNames label
G:=SmallGroup(128,2028);
// by ID
G=gap.SmallGroup(128,2028);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations